Cellular origins of cancer

Chapter 3 presented an extensive stochastic analysis of a two-hit model. In particular we calculated the probability of creating a double-mutant as a function of time, depending on the population size and the relative fitness of the intermediate type. Chapter 4 made the first attempt to apply this model to real-life carcinogenesis, by taking account of specific features of sporadic and familial colorectal cancers. One important consideration which was not included in the analysis so far is the population structure. In Chapters 3 and 4, the population of cells was completely homogeneous with respect to the patterns of mitosis/apoptosis. In other words, cells were only characterized by their "fitness", which was a function of acquired mutations. In some cases, this is not enough to grasp the essential dynamics of the system. An example is the colonic epithelial tissue. There, when talking about the dynamics of cell division and mutations, we may have to take into account the fact that stem cells behave differently from differentiated cells. The analysis which follows can be of importance for one of the fundamental questions in cancer research, namely, from which cells in our body does cancer originate?

To begin, we will briefly describe important aspects of tissue architecture, development, and function. We have to make a distinction between stem cells and differentiated cells. Stem cells have the ability to divide indefinitely. During this process they give rise to differentiated cells which make up the tissue. The differentiated cells perform their function and eventually die. In the context of stem cells, we have to distinguish between embryonic stem cells and adult stem cells. Embryonic stem cells give rise to the organism during development. They are said to be truly multipotent. That is, they can give rise to any tissue in the body (e.g. lung, liver, brain, colon, skin, etc.). Adult stem cells, on the other hand, are thought to be more restricted. That is, they might only be able to give rise to certain tissues. For example, liver stem cells can only differentiate into "committed" liver cells, or colon stem cells can only differentiate into "committed" colon cells. Adult stem cells are thought to be responsible for maintaining and renewing a given tissue. They may divide at a relatively slow rate, or divide only when new tissue cells need to be created (e.g. when already differentiated cells die). Division of adult stem cells is thought to be asymmetric. That is, division gives rise to one stem cell, and one cell which differentiates into a functioning tissue cell.

Consider the colon as an example. The epithelial lining of the colon is made up of many involutions which are called crypts. There are about 107 crypts within a human colon. Each crypt contains stem cells. The exact number of stem cells per crypt is not known; there might be just one stem cell or a small number of them. Upon division, a stem cell gives rise to one stem cell and one cell which embarks on a journey of differentiation. Before this cell is fully differentiated, it divides a certain number of times. A fully differentiated cell lives for about one week. Then it dies and is washed out of the colon. The first malignant change in colon cancer ensures that the differentiated cell does not die after one week. Instead it remains, and this causes an accumulation of abnormal or transformed cells. The inactivation of the tumor suppressor gene APC is responsible for this behavior. The generation of APC-/- cells (or the inactivation of other genes involved in the Wnt pathway) is the first step toward colon cancer [Katoh (2003); Kinzler and Vogelstein (1998); Polakis (1997); Polakis (1999)]. We are faced with an important question. Did the mutation which inactivates the APC gene occur in the stem cells, or in the cells which differentiate?

Talking in more general terms, cancer cells have been shown to have various characteristics in common with stem cells. Fore example, they have the capacity to divide indefinitely. This does not, however, mean that the origin of cancer lies in stem cells. There are two theories. The stem-cell theory, suggests that the first event happens in a stem cell. The de-differentiation theory, claims that it occurs in a (partially)-differentiated cell, thus leading to its de-differentiation, or "immortalization". Experimentally this is a very difficult question, and the debate is ongoing.

Despite this uncertainty, many researches feel that differentiated cells are unimportant for cancer initiation, for the following (quantitative) reason, unrelated to biological evidence. Let us concentrate on colon cancer. It is widely believed that the APC gene is a tumor suppressor gene. That is, the inactivation of both copies is required to confer phenotypic changes

[Kinzler and Vogelstein (1998); Macleod (2000)] (consistent with Knudson's two-hit hypothesis [Knudson (1996)]). Then an obvious question arises: how can the first mutation occur in the migrating compartment, without being washed away? As John Cairns writes [Cairns (2002)], "...there are 256 exponentially multiplying cells that divide twice a day and are being replenished continually by the divisions of a single stem cell, none of these 256 cells will ever be separated from the stem cell by more than eight divisions, and the replication errors made in those eight divisions are destined, of course, to be discarded".

The point of this chapter is to address exactly this issue: will migrating cell mutations be indeed discarded, or is there a chance that they will persist until the second hit comes, which immortalizes the cell and thus initiates dysplasia in the crypt?

5.1 Stem cells, tissue renewal and cancer

The normal functioning of colon relies on the fine-tuned balance of the epithelial cell production, differentiation and death. The regulation of the processes of cell proliferation and shedding occurs at the level of crypts -the folds of colonic epithelium which are continuously renewed by stem cell division. The appearance of dysplastic crypts in the beginning of colorectal cancers is a manifestation of the broken balance between cell division and apoptosis. At the molecular level, it has been shown that the earliest event of sporadic colorectal cancers is the inactivation of the APC gene [Kinzler and Vogelstein (1998); Polakis (1997)], or other genes involved in the Wnt pathway [Katoh (2003); Polakis (1999)]. The APC gene inhibits members of the Wnt signaling pathway, which promote the expression of /3-catenin. In its turn, /3-catenin acts as an enhancer of cell division [Behrens et al. (1998)].

It is widely believed that the relevant target cells for the first mutation are the colonic stem cells, [Bach et al (2000); Fuchs and Segre (2000); Kim and Shibata (2002); Potten et al. (2003); Potten and Loeffler (1990); Winton (2001); Wong et al. (2002)]. The argument usually goes in the following way, see e.g. [Cairns (2002)]. If the first mutation happened in a proliferative daughter cell, it would be washed away before the second hit has a chance to confer a significant phenotypic change. On the other hand, if the first mutation occurs in the "immortal" stem cell, then its mutant progeny will populate the compartment and persist for as long as it takes to accumulate further mutations which give rise to neoplasia.

It is often assumed that the stem cells are located in a niche at the base of the crypt. The "bottom-up" model of colorectal histogenesis [Preston et al. (2003); Wong et al. (2002); Wright and Poulsom (2002)] states that the oncogenic mutations occur in the stem cells at the base of the crypt. A natural consequence of this model is that once such a mutation occurs, the entire crypt will be monoclonally-mutant. There has been some evidence which contradicts this view. Namely, dysplastic cells exhibiting genetic alterations in the APC gene have been found in the upper layer of crypts, whereas cells located at the bottom of the same crypt did not contain such alterations. This led to the "top-down" model [Shih et al. (2001a)]. The two explanations proposed were that (i) the stem cells reside near the top of the crypts, or (ii) transformed cells originate from the stem cells at the base of the crypt, then they passively move upward, after which a cycle of cell proliferation and tissue replacement starts in the top-down direction.

Another explanation has been put forward which suggests that the relevant mutations occur in fully differentiated cells [Fodde et al. (2001b)], or in the proliferative/migrating daughter cells [Lamprecht and Lipkin (2002)]. This is consistent with the stem cells being located at the base of the crypt, as previously thought. At the basis of these models is the idea that a proliferating daughter cell with a silenced APC gene (or otherwise upregulated /3-catenin) will acquire a stem cell phenotypic characteristic, that is, permanence in the crypt. Indeed, it has been shown that the /3-catenin/T cell factor 4 complex constitutes the "master switch" that controls proliferation versus differentiation in healthy and malignant intestinal epithelial cells [van de Wetering et al. (2002)]. Inactivation of the APC gene in a migrating cell could reverse the process of differentiation and trick the cell into thinking that it is "immortal". This would lead to a continued proliferation of this cell which would avoid entering the final differentiation and programmed apoptosis stage.

The following sections will discuss these mechanisms mathematically.

5.2 The basic renewal model

We assume that the stem cells are located in a niche at the base of the crypt. They are characterized by an asymmetric division pattern resulting in one stem cell and one proliferative daughter cell. The latter cells divide and populate the migrating compartment. Cells of the migrating compartment go through a number of symmetric divisions, moving toward the crypt surface. On their way up, they go through stages of differentiation, until the fully differentiated cells stop dividing, reach the crypt surface and get shed into the lumen, to be replaced by new generations of cells coming from the bottom of the crypt. We will refer to the symmetrically dividing, migrating progeny of a stem cell (SC) as "differentiated cells" (DC), keeping in mind that the degree of differentiation increases with the number of divisions that separate the progeny from the stem cell.

Fig. 5.1 The history of one daughter DC. It undergoes 3 rounds of division. The number of cells in the last generation is 8.

Figure 5.1 traces the offspring of one DC created from a SC. Different levels represent consecutive moments of time (or rounds of proliferation). Another interpretation of this figure is spatial: we can think of cells of consecutive generations to be located closer and closer to the top of the crypt. The cells of "age 4" are the closest to the top, and they are shed into the lumen. This can be better seen in Figure 5.2.

The progeny of a single daughter cell is marked by the same letter. The degree of maturation/differentiation is reflected in the intensity of shading: the darker the circle is, the more mature is the cell. The apoptotic cells are presented by dashed circles. At all moments of time, the crypt contains DCs of 4 generations. In the beginning (the leftmost diagram) there are progeny of cells A, B and C, and a newly produced daughter cell, D. After some time (the middle diagram), all of the "oldest" cells (marked with A) have been shed, the cells B and C went through one round of division, advancing their degree of maturation and moving upward, and a daughter

Apoptosis

Apoptosis

Fig. 5.1 The history of one daughter DC. It undergoes 3 rounds of division. The number of cells in the last generation is 8.

Fig. 5.2 Schematic snapshots of one crypt at three moments of time. The stem cell is marked by "S".

cell, E, has been produced. The process of renewal goes on in this way, eventually replacing all DCs in the crypt [Komarova and Wang (2004)].

In Figure 5.2 only one stem cell is shown to repopulate the crypt. In reality, there are several stem cells per crypt, and thus each crypt is a composition of several clones. This can be easily included in our model: if, for example, there are four stem cells in the crypt, then this crypt can be viewed as a "superposition" of four crypts. The general rule that more mature cells are situated closer to the top of the crypt still holds, and the general dynamics of each of the clones is as in Figure 5.2. The only difference is in the numbers that should be used in the model.

Several models of SC dynamics have been designed, [Ro and Rannala (2001); Yatabe et al. (2001)]. In these papers, two main mechanisms of SC reproduction have been proposed. In the deterministic model, each SC divides asymmetrically, and the number of SC is kept constant. In more sophisticated models, each SC has a probability to produce upon division (i) two SCs, (ii) one SC and one DC, or (iii) two DCs. In this model, the number of stem cells fluctuates. The latter model seems to be more realistic. Here, we will use the simpler model, and note that the methods and results developed should remain the same, with minimal changes, if the reproduction model for stem cells is refined.

5.3 Three scenarios

Let us first describe the process of accumulation and spread of mutations in a dynamic crypt. By "mutations" we mean any kind of genetic alteration (a point mutation, a loss of heterozygocity (LOH) event, etc.) which leads to the inactivation of an allele of the APC gene. We will assume that a cell with a single mutation has the same properties as a wild type cell, and a double-mutant has the ability to avoid apoptosis, continue divisions and thus remain and spread in the crypt.

Fig. 5.3 Three scenarios of the emergence of a double mutant, (a), the ss scenario, (b), the sd scenario, (c), the dd scenario. The open circles are wild-type cells, circles with an "x" contain one mutation in the tumor suppressor gene, and double-x's are double mutants.

There are three logical possibilities of accumulation of mutations, see Figure 5.3.

(i) In the ss scenario, Figure 5.3a, a mutation happens in the stem cell. Then, after a few divisions, the entire crypt will consist of mutated cells. At some point, a second mutation occurs in the SC, shortly after which the entire crypt will consist of double mutants. This is the scenario consistent with the "bottom-up" hypothesis. It predicts that the crypt will be monoclonal with respect to double mutations.

(ii) In the sd scenario, Figure 5.3b, again a mutation occurs in the SC which then spreads throughout the crypt. However, the first double-mutant emerges in the proliferating/migrating compartment. This mutant divides and its progeny first spreads in the upward direction. At this point, the lower part of the crypt is monoclonal with respect to one mutation in the APC gene, and the upper part of the crypt is monoclonal with respect to two mutations.

(iii) In the dd scenario, Figure 5.3c, a mutation occurs in one of the migrating daughter cells. The cell divides, its progeny moves in the upward direction, but before it undergoes apoptosis, one of these cells experiences a second hit, creating a double mutant. As a result, the lower part of the crypt consists of wild-type cells, and the upper part is composed of monoclonal double-mutants.

5.4 Mathematical analysis

It is convenient to introduce the quantity I, the total number of division rounds during the life-span of one clone, see Figure 5.1. This number includes one asymmetric division of the stem cell and l—l rounds of symmetric divisions of the DCs (for simplicity we assume that these are synchronized). The total number of progeny of an SC existing at any one time in a crypt, is 2l - 1. We denote N = 2l. For example, in Figure 5.1 we have I = 4, N = 16.

Since the probability of the first hit, pi, is very small, it can be shown that most of the time the population of the crypt will be homogeneous, that is, most of the cells will either be wild-type, or will contain one mutation, see also [Komarova et al. (2003)]. Indeed, if a mutation arises in a DC, it gets washed out in less than I time-steps; in fact, most of the mutants are very short-lived, and only survive for one or two time-steps, because at each moment of time, the majority of the crypt consists of cells only one or two steps away from apoptosis, see Figure 5.1. The frequency with which new mutants are created is Npi, and only about 1/N of these mutants will happen in an SC. So the condition pi <C 1/jV (the homogeneity condition)

guarantees that the crypt contains no mutants most of the time. Unless, of course, a mutation occurs in a SC, in which case in less than I time-units, the entire crypt will consist of mutant cells. Finally, a double mutant may appear, but in this case we assume that the process is over and a dysplastic crypt has been created. Note that the homogeneity condition easily holds for the realistic values, pi ~ 10~7 [Albertini et al. (1990)], and N ~ 103 [Kim and Shibata (2002); Potten and Loeffler (1990)].

Let us call the probability to find the entire crypt consisting of wild-type cells, xq. The probability that the entire crypt consists of cells with a single mutation is x\, and the probability that the crypt is dysplastic (contains one or more double mutants) is x2. The symbols X0, X\ and X2 will be used to denote the corresponding states. Because of the homogeneity condition, we have £o + + £2 ~ 1- ^ double mutant can be created via two major pathways. One pathway includes a fixation of a single mutant. First, a mutation happens in a SC (the rate is pi), after which the entire crypt enters state X\, and then a new mutation occurs (with rate Np2), which brings the crypt to the state X2. Alternatively, a second mutation can occur in a mutant clone which originates in a DC, without a prior fixation of a single mutant. This happens with the rate R, which we now calculate.

Calculation the rate of double mutant generation, for path dd. Let us write down the probability to have at least one mutant such that both mutations happen in the DC (given that no SC mutations have happened). We assume that the mutation rate, pi, is sufficiently small such that the clones can be treated independently (the condition is p\N <C 1), and consider a doubly stochastic process, see also [Iwasa et al. (2004); Moolgavkar et al. (1988)]. We obtain i

To see this, we write down the total rate of primary mutations, Ei=l Pi2i_1 = PiN (this is valid for p\N <C 1). Then each term n = pi2i_1 is a contribution corresponding to the first mutation happening in a DC of generation i. Vi is related to the "secondary" stochastic process which happens in the clone after the first mutation. Inside such a clone, we have _ 2 cell divisions, so that the total probability to get a second hit is

We have i=1

This expression can be calculated by replacing the summation with an integral,

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